Stochastic Well-posed Systems and Well-posedness of Some Stochastic Partial Differential Equations with Boundary Control and Observation

TL;DR

This paper extends the concept of well-posed linear systems to stochastic control systems and demonstrates the well-posedness of stochastic PDEs like heat and Schrödinger equations with boundary control.

Contribution

It introduces a generalized framework for stochastic well-posed systems and proves well-posedness for specific stochastic PDEs with boundary control and observation.

Findings

Well-posedness of stochastic heat equation with boundary control

Well-posedness of stochastic Schrödinger equation with boundary observation

Generalization of well-posed linear systems to stochastic settings

Abstract

We generalize the concept "well-posed linear system" to stochastic linear control systems and study some basic properties of such kind systems. Under our generalized definition, we show the well-posedness of the stochastic heat equation and the stochastic Schr\"odinger equation with suitable boundary control and observation operators, respectively.